In this work we are guided by Lang's conjectures on varieties of general type, and by a question of Serre on covers of abelian varieties. By Kawamata's structure result for ramified covers of abelian varieties [1]}, for every ramified cover \(X\rightarrow A\)  of an abelian variety \(A\)  a number field \(K\) , there is a finite field extension \(L/K\)  and finite étale cover \(X^{\prime }\rightarrow X_L\)  such that \(X^{\prime }\)  dominates a positive-dimensional variety of general type over \(L\) . Assuming Lang's conjecture [2]}, [3]}, it follows that the \(K\) -rational points \(X(K)\)  are not Zariski-dense in \(X\) . Despite Faltings' deep finiteness theorems for integral (and rational) points on subvarieties of abelian varieties [4]}, we are far from proving such desired non-density results for rational points on ramified covers of abelian varieties (even in dimension two) over number fields.
